Abstract
Krylov complexity has been proposed as a diagnostic of chaos in non-integrable lattice and quantum mechanical systems, and if the system is chaotic, Krylov complexity grows exponentially with time. However, when Krylov complexity is applied to quantum field theories, even in free theory, it grows exponentially with time. This exponential growth in free theory is simply due to continuous momentum in non-compact space and has nothing to do with the mass spectrum of theories. Thus by compactifying space sufficiently, exponential growth of Krylov complexity due to continuous momentum can be avoided. In this paper, we propose that the Krylov complexity of operators such as \( \mathcal{O} \) = Tr[FμνFμν] can be an order parameter of confinement/deconfinement transitions in large N quantum field theories on such a compactified space. We explicitly give a prescription of the compactification at finite temperature to distinguish the continuity of spectrum due to momentum and mass spectrum. We then calculate the Krylov complexity of \( \mathcal{N} \) = 4, 0 SU(N) Yang-Mills theories in the large N limit by using holographic analysis of the spectrum and show that the behavior of Krylov complexity reflects the confinement/deconfinement phase transitions through the continuity of mass spectrum.
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References
E.P. Wigner, On a Class of Analytic Functions from the Quantum Theory of Collisions, Annals Math. 53 (1951) 36.
F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].
M.V. Berry and M. Tabor, Level Clustering in the Regular Spectrum, Proc. Roy. Soc. Lond. A 356 (1977) 375.
O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984) 1 [INSPIRE].
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity, JETP 28 (1969) 1200.
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
D.E. Parker et al., A Universal Operator Growth Hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [INSPIRE].
V. Balasubramanian, P. Caputa, J.M. Magan and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106 (2022) 046007 [arXiv:2202.06957] [INSPIRE].
V. Balasubramanian, J.M. Magan and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107 (2023) 126001 [arXiv:2208.08452] [INSPIRE].
J. Erdmenger, S.-K. Jian and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, JHEP 08 (2023) 176 [arXiv:2303.12151] [INSPIRE].
A. Bhattacharyya et al., Krylov complexity and spectral form factor for noisy random matrix models, JHEP 10 (2023) 157 [arXiv:2307.15495] [INSPIRE].
A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104 (2021) L081702 [arXiv:2104.09514] [INSPIRE].
A. Avdoshkin, A. Dymarsky and M. Smolkin, Krylov complexity in quantum field theory, and beyond, arXiv:2212.14429 [INSPIRE].
H.A. Camargo, V. Jahnke, K.-Y. Kim and M. Nishida, Krylov complexity in free and interacting scalar field theories with bounded power spectrum, JHEP 05 (2023) 226 [arXiv:2212.14702] [INSPIRE].
A. Kundu, V. Malvimat and R. Sinha, State dependence of Krylov complexity in 2d CFTs, JHEP 09 (2023) 011 [arXiv:2303.03426] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. B 45 (1950) 255 [INSPIRE].
V.S. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Dynamics, Springer Berlin, Heidelberg, Germany, (1994) [https://doi.org/10.1007/978-3-540-48651-0].
N. Iizuka and J. Polchinski, A Matrix Model for Black Hole Thermalization, JHEP 10 (2008) 028 [arXiv:0801.3657] [INSPIRE].
N. Iizuka and M. Nishida, Krylov complexity in the IP matrix model, JHEP 11 (2023) 065 [arXiv:2306.04805] [INSPIRE].
N. Iizuka and M. Nishida, Krylov complexity in the IP matrix model. Part II, JHEP 11 (2023) 096 [arXiv:2308.07567] [INSPIRE].
J.L.F. Barbón, E. Rabinovici, R. Shir and R. Sinha, On The Evolution Of Operator Complexity Beyond Scrambling, JHEP 10 (2019) 264 [arXiv:1907.05393] [INSPIRE].
D.A. Abanin, W.D. Roeck and F. Huveneers, Exponentially Slow Heating in Periodically Driven Many-Body Systems, Phys. Rev. Lett. 115 (2015) 256803 [arXiv:1507.01474] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
G. Festuccia and H. Liu, Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I, JHEP 04 (2006) 044 [hep-th/0506202] [INSPIRE].
C. Csaki, H. Ooguri, Y. Oz and J. Terning, Glueball mass spectrum from supergravity, JHEP 01 (1999) 017 [hep-th/9806021] [INSPIRE].
R. de Mello Koch, A. Jevicki, M. Mihailescu and J.P. Nunes, Evaluation of glueball masses from supergravity, Phys. Rev. D 58 (1998) 105009 [hep-th/9806125] [INSPIRE].
R.C. Brower, S.D. Mathur and C.-I. Tan, Discrete spectrum of the graviton in the AdS5 black hole background, Nucl. Phys. B 574 (2000) 219 [hep-th/9908196] [INSPIRE].
R.C. Brower, S.D. Mathur and C.-I. Tan, Glueball spectrum for QCD from AdS supergravity duality, Nucl. Phys. B 587 (2000) 249 [hep-th/0003115] [INSPIRE].
D.S. Lubinsky, H.N. Mhaskar and E.B. Saff, A proof of Freud’s conjecture for exponential weights, Constructive Approx. 4 (1988) 65.
Acknowledgments
The works of TA and NI were supported in part by JSPS KAKENHI Grant Number 21J20906(TA), 18K03619(NI). The work of NI is also supported by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A “Extreme Universe” No. 21H05184. M.N. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2023-00245035).
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Anegawa, T., Iizuka, N. & Nishida, M. Krylov complexity as an order parameter for deconfinement phase transitions at large N. J. High Energ. Phys. 2024, 119 (2024). https://doi.org/10.1007/JHEP04(2024)119
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DOI: https://doi.org/10.1007/JHEP04(2024)119